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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 6525j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6525.k2 | 6525j1 | \([1, -1, 0, -162, -329]\) | \(5177717/2349\) | \(214052625\) | \([2]\) | \(1536\) | \(0.29358\) | \(\Gamma_0(N)\)-optimal |
| 6525.k1 | 6525j2 | \([1, -1, 0, -2187, -38804]\) | \(12698260037/7569\) | \(689725125\) | \([2]\) | \(3072\) | \(0.64015\) |
Rank
sage: E.rank()
The elliptic curves in class 6525j have rank \(1\).
Complex multiplication
The elliptic curves in class 6525j do not have complex multiplication.Modular form 6525.2.a.j
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
